We propose a hierarchical Bayesian nonparametric mixture model for
clustering when some of the covariates are assumed to be of varying relevance to
the clustering problem. This can be thought of as an issue in variable selection
for unsupervised learning. We demonstrate that by de¯ning a hierarchical pop-
ulation based nonparametric prior on the cluster locations scaled by the inverse
covariance matrices of the likelihood we arrive at a `sparsity prior' representation
which admits a conditionally conjugate prior. This allows us to perform full Gibbs
sampling to obtain posterior distributions over parameters of interest including an
explicit measure of each covariate's relevance and a distribution over the number
of potential clusters present in the data. This also allows for individual cluster
speci¯c variable selection. We demonstrate improved inference on a number of
canonical problems.